{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T02:57:24Z","timestamp":1772765844564,"version":"3.50.1"},"reference-count":24,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,12,22]],"date-time":"2023-12-22T00:00:00Z","timestamp":1703203200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia","award":["5120"],"award-info":[{"award-number":["5120"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this work, the geometric nature of solutions to two second-order differential equations, zy\u2032\u2032(z)+a(z)y\u2032(z)+b(z)y(z)=0 and z2y\u2032\u2032(z)+a(z)y\u2032(z)+b(z)y(z)=d(z), is studied. Here, a(z), b(z), and d(z) are analytic functions defined on the unit disc. Using differential subordination, we established that the normalized solution F(z) (with F(0) = 1) of above differential equations maps the unit disc to the domain bounded by the leminscate curve 1+z. We construct several examples by the judicious choice of a(z), b(z), and d(z). The examples include Bessel functions, Struve functions, the Bessel\u2013Sturve kernel, confluent hypergeometric functions, and many other special functions. We also established a connection with the nephroid domain. Directly using subordination, we construct functions that are subordinated by a nephroid function. Two open problems are also suggested in the conclusion.<\/jats:p>","DOI":"10.3390\/sym16010019","type":"journal-article","created":{"date-parts":[[2023,12,22]],"date-time":"2023-12-22T08:53:01Z","timestamp":1703235181000},"page":"19","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Geometric Nature of Special Functions on Domain Enclosed by Nephroid and Leminscate Curve"],"prefix":"10.3390","volume":"16","author":[{"given":"Reem","family":"Alzahrani","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, King Faisal University, Al Ahsa 31982, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4540-1601","authenticated-orcid":false,"given":"Saiful R.","family":"Mondal","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, King Faisal University, Al Ahsa 31982, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Mondal, S.R. (2022). On Lemniscate Starlikeness of the Solution of General Differential Equations. Mathematics, 10.","DOI":"10.3390\/math10183254"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Mondal, S.R. (2022). Mapping Properties of Associate Laguerre Polynomials in Leminiscate, Exponential and Nephroid Domain. Symmetry, 14.","DOI":"10.3390\/sym14112303"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Mondal, S.R. (2022). Subordination Involving Regular Coulomb Wave Functions. Symmetry, 14.","DOI":"10.3390\/sym14051007"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Alarifi, N.M., and Mondal, S.R. (2022). On Geometric Properties of Bessel\u2013Sturve Kernel Functions in Unit Disc. 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Anal."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/1\/19\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:40:25Z","timestamp":1760132425000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/1\/19"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,22]]},"references-count":24,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,1]]}},"alternative-id":["sym16010019"],"URL":"https:\/\/doi.org\/10.3390\/sym16010019","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,12,22]]}}}